VPV Identities related to ${\mathbf{x}}^{\mathbf{y}}={\mathbf{y}}^{\mathbf{x}}$ and ${\mathbf{x}}^{\mathbf{y}}{\mathbf{y}}^{\mathbf{x}}={\mathbf{v}}^{\mathbf{w}}{\mathbf{w}}^{\mathbf{v}}$

Abstract

We cover rational and integer solutions for the equations ${\mathbf{x}}^{\mathbf{y}}={\mathbf{y}}^{\mathbf{x}}$ and ${\mathbf{x}}^{\mathbf{y}}{\mathbf{y}}^{\mathbf{x}}={\mathbf{v}}^{\mathbf{w}}{\mathbf{w}}^{\mathbf{v}}$. The former equation solutions go back to Euler, and the latter equation solutions appear to be new. Another definitely new related topic is application of VPV identities to give transforms of infinite products from our solutions. The present paper is essentially chapter 28 of the author's book appearing in June 2024.

6 pages. Chapter 28 of the book: CAMPBELL, G.B. Vector Partitions, Visible Points, and Ramanujan Functions, CRC Press, Taylor and Francis Group, Boca Raton, London, New York, A Chapman & Hall Book, ISBN: 978-1-032-00366-5 (hbk), ISBN: 978-1-032-00432-7 (pbk), ISBN: 978-1-003-17415-8 (ebk), to appear, June 2024

References in corpus (2)

• VPV Identities related to ${\mathbf{x}}^{\mathbf{y}}={\mathbf{y}}^{\mathbf{x}}$ and ${\mathbf{x}}^{\mathbf{y}}{\mathbf{y}}^{\mathbf{x}}={\mathbf{v}}^{\mathbf{w}}{\mathbf{w}}^{\mathbf{v}}$
• Rogers-Ramanujan identities in Statistical Mechanics

Cited by in corpus (3)

• VPV Identities related to ${\mathbf{x}}^{\mathbf{y}}={\mathbf{y}}^{\mathbf{x}}$ and ${\mathbf{x}}^{\mathbf{y}}{\mathbf{y}}^{\mathbf{x}}={\mathbf{v}}^{\mathbf{w}}{\mathbf{w}}^{\mathbf{v}}$
• Rogers-Ramanujan identities in Statistical Mechanics
• Boundary critical phenomena in the quantum Ashkin-Teller model